2Q. A flow field is defined by u = 2y, v=xy. Derive expressions for the x and y components of acceleration. Find the magnitude of the velocity and acceleration at the point (2,3). Specify units in terms of L and T.
2Q. A flow field is defined by u = 2y, v=xy. Derive expressions for the x and y components of acceleration. Find the magnitude of the velocity and acceleration at the point (2,3). Specify units in terms of L and T.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![**Problem Statement:**
A flow field is defined by \( u = 2y \), \( v = xy \). Derive expressions for the \( x \) and \( y \) components of acceleration. Find the magnitude of the velocity and acceleration at the point (2,3). Specify units in terms of \( L \) and \( T \).
**Solution:**
1. **Velocity Components:**
- The velocity components are given as:
\[
u = 2y
\]
\[
v = xy
\]
2. **Acceleration Components:**
- The \( x \) component of acceleration, \( a_x \), is given by:
\[
a_x = \frac{du}{dt} = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}
\]
Since \( u = 2y \), \(\frac{\partial u}{\partial x} = 0\) and \(\frac{\partial u}{\partial y} = 2\).
- The \( y \) component of acceleration, \( a_y \), is given by:
\[
a_y = \frac{dv}{dt} = \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}
\]
Since \( v = xy \), \(\frac{\partial v}{\partial x} = y\) and \(\frac{\partial v}{\partial y} = x\).
3. **Calculate Specific Values:**
- Evaluate both components of acceleration and the magnitude of velocity at the point (2,3).
- **Magnitude of Velocity:**
\[
\text{Velocity magnitude} = \sqrt{u^2 + v^2}
\]
- **Magnitude of Acceleration:**
\[
\text{Acceleration magnitude} = \sqrt{a_x^2 + a_y^2}
\]
4. **Units:**
- Velocity is measured in \( L/T \).
- Acceleration is measured in \( L/T^2 \).
By applying the given definitions and calculations, one can derive the necessary expressions for acceleration and determine the required magnitudes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93c89597-7ce2-4451-bd61-6716aa1c2201%2Fe6ac2190-1a6f-4603-a6fc-bce646a5be42%2Fjxgko7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A flow field is defined by \( u = 2y \), \( v = xy \). Derive expressions for the \( x \) and \( y \) components of acceleration. Find the magnitude of the velocity and acceleration at the point (2,3). Specify units in terms of \( L \) and \( T \).
**Solution:**
1. **Velocity Components:**
- The velocity components are given as:
\[
u = 2y
\]
\[
v = xy
\]
2. **Acceleration Components:**
- The \( x \) component of acceleration, \( a_x \), is given by:
\[
a_x = \frac{du}{dt} = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}
\]
Since \( u = 2y \), \(\frac{\partial u}{\partial x} = 0\) and \(\frac{\partial u}{\partial y} = 2\).
- The \( y \) component of acceleration, \( a_y \), is given by:
\[
a_y = \frac{dv}{dt} = \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}
\]
Since \( v = xy \), \(\frac{\partial v}{\partial x} = y\) and \(\frac{\partial v}{\partial y} = x\).
3. **Calculate Specific Values:**
- Evaluate both components of acceleration and the magnitude of velocity at the point (2,3).
- **Magnitude of Velocity:**
\[
\text{Velocity magnitude} = \sqrt{u^2 + v^2}
\]
- **Magnitude of Acceleration:**
\[
\text{Acceleration magnitude} = \sqrt{a_x^2 + a_y^2}
\]
4. **Units:**
- Velocity is measured in \( L/T \).
- Acceleration is measured in \( L/T^2 \).
By applying the given definitions and calculations, one can derive the necessary expressions for acceleration and determine the required magnitudes
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